Generalization Error Analysis of Deep Physical Models With Latent Variables Trained on Trajectory Data

Yuhan Chen, Takashi Matsubara, Takaharu Yaguchi

Keywords: Hamiltonian Neural Networks, Generalization Error, AI for Science

Abstract: In this paper, we investigate the generalization error of deep physical models with latent variables. Deep physical models, such as Hamiltonian Neural Networks, are neural network models for learning equations of motion from observational data of physical phenomena and have attracted much attention in recent years. In particular, in such cases, the data are not completely random, but rather given as random trajectories. We provide an error bound for the case where the training data are given in such a way. Our results show that it is important to collect data from many trajectories, rather than simply collecting a large number of data, to improve generalization performance. In addition, an important application of the combination of deep physics models with latent variables is the interpolation of images from videos while preserving the laws of physics, such as the energy conservation law. However, when the frame interval of the video is large, it can be difficult to preserve the laws of physics. In this paper, we show that it is possible to interpolate the images from videos so that the laws of physics are preserved, provided that the generalization error is sufficiently small.

Primary Area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)

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Submission Number: 8931